The Existence of Solutions to Sturm-Liouville Boundary Value Problems with Laplacian-like Operator

Abstract The existence of infinitely many solutions to Sturm-Liouville boundary value problem with aLaplacian-like operator is studied by applying generalized polar coordinates.     

Half-inverse problem for diffusion operators on the finite interval

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on , then only one spectrum is sufficient to determine q(x) on the interval for diffusion operator.     

参数边界条件下奇型Sturm-Liouville算子的半逆问题

Sturm-Liouville算子的半逆问题讨论由一组谱和半区间上势函数唯一确定整个区间上势函数q(x).本文利用Koyunbakan和Panakhov的方法和[13]的结论,讨论(0,π)上的奇型Sturm-Liouville问题满足-y″+[q(x)-1/4sin2x]y=λy,参数边界条件y(0,λ)=0或y′(0,λ)-hy(0,λ)=0和y′(π,λ)+(aλ+b)y(π,λ)=0,证明一组谱和(π/2,π)上的势函数q(x)唯一确定(0,π)上的势函数q(x).     

具有算子系数的微分算子新的Ambarzumyan定理(英文)

有限区间上具有Neumann边界条件的Sturm-Liouville问题的谱可以唯一确定势函数,这就是经典的Ambarzumyan定理.本文将经典的Ambarzumyan定理推广到有限区间上具有算子系数的二阶与四阶微分算子中.     

φ凹(-φ凸)算子不动点定理及其应用

讨论了φ凹(-φ凸)算子,得到了φ凹增(-φ凸减)算子不动点存在唯一性结果,并且给出了收敛到该不动点的迭代序列.该结果去掉了以往文献中的φ→0(t→0~+)这一条件,从而改进和推广了相关结果.作为应用,给出了一类的Sturm-Liouville边值问题的正解的存在唯一性结果.     

On the resolvent of singular Sturm-Liouville operators with transmission conditions

In this article, we investigate the resolvent operator of singular Sturm-Liouville problem with transmission conditions. We obtain integral representations for the resolvent of this operator in terms of the spectral function. Later, we discuss some properties of the resolvent operator, such as Hilbert-Schmidt kernel property, compactness. Finally, we give a formula in terms of the spectral function for the Weyl-Titchmarsh function of this problem.     

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

奇型Sturm-Liouville微分算子的限界自伴扩张

本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件.     

On the similarity of a <Emphasis Type="Italic">J</Emphasis>-nonnegative Sturm-Liouville operator to a self-adjoint operator

In terms of Weyl-Titchmarsh m-functions, we obtain a new necessary condition for an indefinite Sturm-Liouville operator to be similar to a self-adjoint operator. This condition is used to construct examples of J-nonnegative Sturm-Liouville operators with singular critical point zero.     

Invariant transformations for the Sturm-Liouville operator

In this paper, we consider the Sturm-Liouville operator on a finite interval. For particular boundary conditions, a group of invariant transformations that preserve the operator spectrum is constructed. This result allows us to reconsider some old problems for the Sturm-Liouville operator. In particular, the influence of the group of transformations on the inverse problem is discussed. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 25–54.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.     

Spectral analysis of indefinite Sturm-Liouville operators

For a class of indefinite J-nonnegative Sturm-Liouville operators, we present a criterion of similarity to a self-adjoint operator. This criterion is formulated in terms of Weyl-Titchmarsh m-functions. Moreover, using this result, we obtain a criterion, as well as simple sufficient conditions, formulated in terms of the coefficients of a given Sturm-Liouville operator.     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 +q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.     

一类带不定权函数的奇型Sturm-Liouville算子的局部可定性(英文)

研究一类带不定权函数的奇型Sturm-Liouville算子,给出相应自伴算子在无穷点邻域的局部可定性.     

奇型Sturm-Liouville算子的半逆问题

奇型Sturm-Liouville问题的势函数q(x)由两组谱唯一确定的,Sturm-Liouville算子的半逆问题讨论由一组谱和一半势函数唯一确定势函数q(x).本文利用Koyunbakan和Panakhov的方法,证明一组谱和(π/2,π)上的势函数q(x)唯一确定(0,π)上Sturm-Liouville方程含有奇型的1sin2x的势函数q(x).     

Spectrum of a Jacobi matrix with exponentially growing matrix elements

A Jacobi matrix with an exponential growth of its elements and the corresponding symmetric operator are considered. It is proved that the eigenvalue problem for some self-adjoint extension of this operator in some Hilbert space is equivalent to the eigenvalue problem of the Sturm-Liouville operator with a discrete self-similar weight. An asymptotic formula for the distribution of eigenvalues is obtained.     

On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 517–524, April, 1995.     

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrödinger operator.